Relations and Functions

Definitions and Key Concepts

Understanding the distinction between relations and functions is foundational in Business Calculus, as functions model various business scenarios mathematically.

• Relation: A relation is a connection between two sets of values, typically represented as ordered pairs (x, y).

• Function: A function is a special type of relation where each input (x-value) has at most one output (y-value).

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Example: The set {(1,2), (2,3), (3,4)} is a function, but {(1,2), (1,3), (2,4)} is not, since the input 1 maps to two outputs.

Inputs and Outputs

• Inputs (x): The independent variable, often representing time, quantity, or other business metrics.

• Outputs (y): The dependent variable, representing outcomes such as profit, cost, or revenue.

Practice Problems

• Given the relation {(-3,5), (0,2), (3,5)}, identify inputs and outputs and determine if it is a function.

• Given the relation {(2,5), (0,2), (2,9)}, identify inputs and outputs and determine if it is a function.

• Determine which of several graphs represent functions using the vertical line test.

Domain and Range

Definitions

Domain and range are essential for describing the set of possible inputs and outputs for a function, which is crucial for modeling constraints in business applications.

• Domain: The set of all allowed input values (x-values) for a function.

• Range: The set of all allowed output values (y-values) for a function.

Finding Domain and Range from Graphs

• To find the domain, project the graph onto the x-axis and identify the interval covered.

• To find the range, project the graph onto the y-axis and identify the interval covered.

Notation

• Interval Notation: Uses brackets [ ] for including endpoints and parentheses ( ) for excluding endpoints. Example: means x ranges from -4 (included) to 5 (not included).

• Set Builder Notation: Describes the set using inequalities. Example: .

Example

• Given a graph, determine the domain and range and express the answer using interval notation.

Piecewise Functions

Definition and Structure

Piecewise functions are used to model situations where a rule or formula changes depending on the input value, which is common in business scenarios such as tiered pricing or tax brackets.

• Piecewise Function: A single function made up of multiple equations, each applying to a different interval of the domain.

• If the function values at the boundaries between pieces do not match, the function has a jump (discontinuity).

General Form

• Example:

Evaluating Piecewise Functions

• To evaluate , determine which interval a falls into and use the corresponding formula.

• Example: For , , .

Graphing Piecewise Functions

• Graph each piece on its respective interval, paying attention to open or closed endpoints.

• Indicate jumps or discontinuities where the formulas change.

Practice Problems

• Given a piecewise function, evaluate at specific points (e.g., , ).

• Graph a piecewise function given its definition.

Summary Table: Function Properties

Additional info: These foundational concepts are essential for understanding more advanced topics in Business Calculus, such as optimization, modeling, and interpreting real-world business scenarios mathematically.